Now, if you start with $3 and Bob starts with $1, the odds change.

You have a 75% chance of winning if you start at $3, and Bob starts at $1.

Those questions were pretty straightforward, and could be guessed just by using logic.

Another Example Problem

You're in a cave, and you have three doors. In one door, you walk around for two days, before returning to your original location. In another door, you walk around for three days, before returning back to the cave. In the last door, you walk for four days, and you leave. What is the average number of days you stay in the cave? (note: assume the doors are random each time. Otherwise, you could go in the first, then the second, then the third)

Using the same method as before, you are able to calculate the average number of days.

For the first two doors, it's 2 + e, and 3 + e, respectively, because you return to your original location, and the expected number of days hasn't been affected.

e = 1/3 * (2 + e) + 1/3 * (3 + e) + 1/3 (4)

e = 5/3 + 2/3e + 4/3e = 3 + 2/3e1/3e = 3e = 9

So, if you're in that situation, you can expected to be in the cave for an average of 9 days.

A Third Example Problem

You have a single bacterium. It has a 1/4 chance of tripling, a 1/4 chance of doubling, a 1/4 chance of staying the same, and a 1/4 chance of dying. What's the chance the colony of bacteria dies off completely?

Each bacterium can either die, stay the same then die, double then die, triple then die, and etc. So you must take into consideration that the situation returns to the original place, just there are more of them. So, the chance of three bacteria dying is p^3, two- p^2, one- p, and etc. For three bacteria, it's p^3, because each bacterium is has a chance p of dying.

The probability can't equal 1, because the bacteria can keep doubling or tripling forever, so that's an extraneous root. Now, the factors of the quadratic are -1±√2, and -1-√2 is negative, so that's also extraneous. So, our answer is -1+√2, or √2 - 1 or ~0.414. Therefore, there's a 41.4% chance that the bacteria die off completely.